 Good. So, the second speaker of the session, which is a pleasure to have, is Axel Cortescubero. And he will talk about GHD as first order in the thermodynamic form factor expansion. A, thermodynamic form factor expansion. Please. So first, sorry, there's like a construction next door. I'm in my house, so there might be noise. And yeah, so thanks for the invitation for organizers. All right, so let's just start. So I will focus in particular. One plus one the integral quantum field theories, because they are particularly simple kind of integral models. This is my one slide summary of quantum field theory. So what we have is a space time continuum is in a continuing theory. And the things we want to observe for correlation functions. So here is a two point function where we have one operator inserted one point and one. And then you look at things like correlation functions, which are rather function of these two operators issues and the separations between the two operators. And this is the picture in a boring old vacuum. This would be very cool to do in the 80s. Okay, so there's a general strategy to compute this kind of correlators in a integral field theory, which is a form factor expansion. So what you do is you insert between the two operators complete set of eigenstates of your theory. And there's a few nice things afforded to you by your relativistic invariance like you can parameterize the X and T dependence in this nice relativistic way, depending on the momentum and energy of this eigenstates. So in a relativistic integral field theory, you know what these eigenstates look like because what you have is a number of particle states on top of vacuum so you can have any number n of particles and you parameterize each particle by the rapidity, which is a way to parameterize their energy and momentum in a relativistic way. Here I'm focusing on a simple, particularly simple example where you just have one kind of particle with one mass and Anyway, so you defined this. This is what is defined as a form factor, which is the matrix element of your field with a given particle states and the vacuum. And the point is, if you know a handful of these four factors, you can approximate correlation function quite simply. So, yes, the first few excitations on top of the vacuum are enough for if you're interested in the long distance properties of the correlation. So, for instance, here for the two point function, the one particle form factor already tells you a lot. So here you put only the one particle state on top of the vacuum. And from this you can compute long distance approximation that you can do a stationary phase approximation and this integral over the rapidity of that one particle. And that tells you that generically a massive integrable field theory has some exponential behavior in the two part function where this either decay or oscillatory depends on whether the separation is time like or space like. So here are a few nice properties of form factors in particular that we will need in this talk. So this is a story for my rustic pictures. There is a worldwide pandemic. So this is what this is what we deserve now. Yeah, so yeah, so this is a nice way of drawing a form factor where you say this is the operator and I have my incoming state with n particles coming in. And here's my outgoing state where I have the vacuum. So this property in our realistic theories that I can from this one obtain a form factor with one outgoing particle by using crossing symmetry you turn one incoming into one outgoing particle and that amounts to shifting the rapidity. Another point is that you have annihilation polls, which basically means that an outgoing particle can annihilate with one of the incoming particles, and this is reflected in the fact that there is a simple poll in the form factor when this rapidity coincides with one of these rapidities and residue of the form factor at that location is proportional to the form factor without those two particles. So those particles having annihilated. And that gives you a sort of a recurrence relation that if you go to that particular point, you can relate the form factor with the one with two fewer particles. And you can do the same with this one also have has polls, a particular values of their business. So now let's talk about GHD, I were here. So this is my interpretation of GHD what it's good for is, so you have some effective theory that allows you to compute correlation functions. So you want to compute a correlation function now on some fancy excited state. So here I call it role, where is some thermodynamic state which has a finite energy density. And the nice point is that this state can also break translation in variants. And that's when GHD is particularly useful. For instance, so this is a kind of state we will examine later on, which you can look at our excited state which looks like generalized keeps example but with a kind of position dependent chemical potentials that makes it a continuous state. And, yeah, so GHD allows you to compute nice things and we are excited states like this, instead of the vacuum which is what you put. So, how GHD works with another rustic picture. The approach is basically you take your space time, and you cut it off into little cells, which are large in a sense that you can do a average of the pills within one cell here. The main assumption in GHD is that you say, okay, so if I look at an operator within one cell, the expectation value here in this cell looks sort of like a normal GD expectation value, but with a chemical set of chemical rich, but with a position dependent. So GHD is basically gives you a way to find an evolution equation for these chemical potentials and once you have that you can get your local expectation values. But this is an assumption in GHD, which is one of the things we want to do here is find this from very brute, very stupid form factors and arrive at this thing instead of making this nice educated assumption. So here are the two goals of what we want to do here. GHD especially shines when you have in homogeneous excited states, but it also has some nice insights for homogeneous states. So there's a particularly nice prediction from this paper for a two point function in a homogeneous but highly excited state. I'll show you what this prediction is later but we will match the prediction from GHD with just doing our form factors. Here there's a spoiler that you see that I have scaled this two point function with the linearly with the separation in time between the two operators because the prediction from GHD gives you a correlator that decays as one over T and you see this is different from the ground state correlators that I showed you before that decay exponentially. So something must happen differently between our ground state intuition that you get exponential decay and this GHD prediction. And then the other thing we want to get to is So we want to compute using form factors one point function in a homogeneous state. So two point functions in homogeneous one point. So let's stop. So this review of thermodynamics what these excited states look like the homogeneous ones. Basically, to say you have a thermodynamic larger number of particles so you have n particles in your in your state. That number grows with the system size and you keep it you keep the density fixed so there's a thermodynamic background and you can further characterize this background by saying the probability that you will find that background a certain particle so you can specify the distribution of this. For instance, you can calculate the energy of this thermodynamic ground state by basically adding over the energy of each particle in this background, governed by this distribution. And I use the ground state loosely because as we'll see. It's not a ground state because you can go to energy lower than that energy as well as higher. So now what we want to do is look at form factors, which are the particle excitations which leave above this thermodynamic background, instead of above the vacuum. So we want to look as we want to study the excitations on top of this thermodynamic state which looks something like this. So this is a particular state that realizes this distribution, and then if you want you can add a couple more particles on top of that state. Now an important thing is that when you add a certain number of particles on top of that state that brings some of the application to the background particles. You can also add a particle with a prime that is not exactly the same background state, but you shifted it by adding more particles. And then the point is that you can also now remove particles from the background, which in this relativistic case is equivalent to adding a particle with a shape the probability like this. So this is different now from the ground state so you didn't have this. You can add that in the ground state because you can remove particles from the ground state. So now we want to look at what we did for the ground state to look at the first form factors that contribute to a two point function. So here is a quick study of what are the low lying excitations on top of this thermodynamic background. So this is something similar to the ground state. So I say so this is my form factor defined on this thermodynamic state row where I say I have the thermodynamic state my operator and the thermodynamic state with one particle on top. So the point is because it's very similar to the one particle form factor from before it actually leads to very qualitatively similar exponential decay behavior. So this is not what we want to reach a GST. Similarly, you could take a one whole form factor where you just remove one particle from the form factor but that will also be to an exponential behavior. So these these two contributions actually are so bleeding they don't contribute to this generalized hydrodynamics regime. So that's how actually the relevant low lying excitation is the one particle whole pair. You can see this is kind of more important than the case slower than these contributions because this one particle whole pair excitation is gapless. So the point is you can create the particle on a whole pair with almost opposite energy that almost cancel each other with a little bit of energy as well. So now what is what we have to do is just straightforward calculated this guy so this is what we want to calculate in our field theory the one particle whole pair on top of some thermodynamic background. Actually, we won't do all of this but we only need to do a simpler thing because we are only interested in the long distance limit of the correlation function. So, when we put this guy in a correlation function we're going to integrate over rapidities and we're going to do a stationary phase approximation. So that leads to the fact that we only need this limit of the front factor where the two particles are rapidities close to each other, the particle on the whole. So, here's the, so this is the whole paper is calculating this thing where I won't go through the gory details of that calculation but just a general strategy. Just consider the finite volume revalorization of this guy, which looks like this. So if you go at finite volume, you can write this in terms of normal form factors with a bunch of particles where that bunch of particles follows the distribution of raw data. So, yeah, so here this is a normal form factor on top of the vacuum. But here so this is the set of rapidities corresponding to your background thermodynamic background state in the outgoing state, and this is your. So I'm talking about this set of particles and this set of particles. So here where this set of particles gets shifted because I added extra excitation so this get a shift. So here I have background particles background particles plus a shift, which the point is that we are completely able to calculate this shift on each of these background particles and it depends on the rapidity of the additional particles and we can work out very carefully what this thing goes to, you know, this nice limits. The price this shift is proportional to this cap so as I take this kappa to zero, all of these sheets will also go to zero. Now, I told you there are annihilation poles whenever some data equals another data plus Pi I so this expression is going to have a bunch of poles is all poles, which looks scary but it's actually nice because that tells you a lot about the structure so also all of these particles are about to annihilate with all of these particles when I take this kappa to zero and this will give you a nicer recurrence relation between this form factor and the one with two fewer particles and then that's also related to fewer particles and pure particles so in the end the expression looks like this where you have all of our form factors with different number of particles where this corresponds to this recurrence relation that this form factor contains all the form factors with fewer and fewer particles and the important point here is that so here we have a fancy form factor we define on top of this thermodynamic background but the form factors in each side are not fancy form factors these are the standard form factors on top of the bucket and this denotes that is a connected part of that form faction which good for you if you know what that is but I won't stop and the point is these are all in principle known all stuff from the 80s that you know how to complete this form factors and knowing this you can compute this address form factors now you take this two part a particle whole form factor put it in your correlator very simply like a couple of lines and you get this two point correlation function so here is just a contribution from the one particle whole pair form factor which in turn is a in turn is a sum over an infinite number of standard form factors and the point is this prediction this computation agrees exactly with the prediction from GHD from that paper by Benjamin that's that for that so so you do the firm the first leading form factor in this simple limit where the rapidities are equal and you recover the prediction from GHD now moving on a bit quickly so so something else that can be done is to look at one point function when you have an inhomogeneous state for instance this state that I described before described by a position dependent chemical potentials so the strategy here is so we want to compute this just with the form factors stupidly without making typical assumptions from GHD so the strategy is so if you assume this this space dependent is slow enough then that you can kind of split this guy into cells where it's European just arbitrary X dependent you have piecewise constant chemical potentials yeah basically you can break this guy into into blocks of size L where does the size of the cell and so there's then you can kind of redefine these chemical potentials of each block by extracting one one overall chemical potential which doesn't depend on the cell you're in so this is arbitrary any chemical potential that I want I will fix it later and the point of this is to kind of extract an overall chemical potential here such that this guy looks like many point functions so here you have the GD expectation value of a bunch of operators one from each cell which corresponds to each of these cells here and this approximation holds in the limit of large cells and what we want to do is to choose the most convenient arbitrary chemical potential that makes this expression the simplest and now this correlation functions we can do with form factors important point is that not all the operators here are non trivially correlated to each other because the point is that this many point function looks like this you have your operator and here you have all your cells in your initial state and the point is that the different cells are not correlated to each other in this leading term we want to calculate because kind of they are outside of each other's life so the correlation between the two of them is the case exponentially while the correlation between this cell and your operator is within the light cone so you only need to evaluate the correlation of the operator between the operator and each cell within the light cone of that operator so in the end your one point function looks like this where you have first just a zero point function of that operator and then corrections from the two point function between your operator and each of these cells in the initial state and the point is these are two point functions that we know how to calculate this is exactly as the one from the previous result like homogeneous correlation function with this set of betas I am yet to choose so we know these guys how to calculate them we can calculate them but the point that makes even nicer is that I am still free to choose this arbitrary betas there exists there is a single simplest choice of this beta which makes kind of so because I know the expression of all these two point functions I can see from that expression that I can make them all vanish at once by choosing a particularly judicious set of chemical potentials and then that happens to be exactly the one that is predicted by the question motions of GHD so I do that I choose this all of these terms vanish I just remain with this one point function in this state and this is the prediction from generalized hydrodynamics derived without assuming you know so that's also yeah so one has access to higher corrections here if one is so inclined by including more fun factors and grinding through it I was not so inclined so there's a future list of things that one would like to do so I looked at homogeneous two point function and inhomogeneous one point function naturally we would like to look at inhomogeneous two point functions for which there are GHD predictions but we haven't gotten there yet another thing is we would like to calculate corrections like this that go beyond the standard GHD predictions and perhaps we can compare with new methods that are coming up recently to go beyond the GHD like for example the Paul I think will talk about quantum corrections to GHD Jacopo might talk about diffusive GHD and so on yeah so there's also you can take classical and or non-relativistic limits of this theory and there you can really compare with a lot of nice doable numerics and things so I don't always have the attention of so many physicists around the world so I would like to use this opportunity to leave you with a much more important and timely problem in physics which is this okay so thanks Axel thanks for the very interesting talk and also for the last slide are there any quick questions I'm sure there are many because it was very interesting but quick for me please Jacopo hi Axel so GHD basically is the saddle point of correlation functions right I mean you obtain GHD if you take saddle points of your correlation functions like large T yes yes basically everything that goes beyond GHD is by not taking the saddle points or there are also other saddle points that you can take that are not included in GHD so to my understanding of this result so the the point function here I wrote so this is the leading contribution and I wrote here plus non-standard GHD corrections so this comes from two sources so one is that so this is from the one particle whole term but not only that this this is in this limit of equal rapidity which comes from stationary phase approximation now stationary phase approximation has corrections which come at higher order in T so this contribute here so there's from the same particle whole form factor corrections but there's also corrections from higher form factors with more particle holes and those completed but the nice thing is you can show those are higher so but if you take like for example the two particle whole pair form factor and do stationary phase approximation the leading contribution is 1 over T square and you can show that yes but the question was more like if you can find also a saddle I mean I mean because now you're taking stationary phase approximation around certain minima right of dispersion relation but maybe there are other minima that are not included in the because these are always limit of zero momentum which are like adrodynamic states but I mean I don't have knowledge of something like that not that I have seen something like that other very quick questions I have one please go ahead hi this is very interesting the question is have you tried to go beyond that point you know that you can compute arbitrary correction after that point have you thought about it or yeah first nice room you have in space like yeah yeah so yes actually we're kind of working on that so one current project we have and we're fighting about this for a while is going from this form factor to the general going from this to this is a project we've been talking about for a while and it's not that simple but I think we'll do it in the near future and once you have the general form factor for any for any two rapidities then you can compute corrections to that so then you can yeah thank you thank you very much